Abstract
We establish some relations between the class of truth-equational logics, the class of assertional logics, other classes in the Leibniz hierarchy, and the classes in the Frege hierarchy. We argue that the class of assertional logics belongs properly in the Leibniz hierarchy. We give two new characterizations of truth-equational logics in terms of their full generalized models, and use them to obtain further results on the internal structure of the Frege hierarchy and on the relations between the two hierarchies. Some of these results and several counterexamples contribute to answer a few open problems in abstract algebraic logic, and open a new one.
Dedicated to Professor Don Pigozzi
on the occasion of his 80th birthday
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Acknowledgements
The authors acknowledge with thanks the following support: grant SFRH/BD/79581/2011 from FCT of the government of Portugal, research grant 2014SGR-788 from the government of Catalonia, research project MTM2011-25747 from the government of Spain, which includes FEDER funds from the European Union, project GA13-14654S of the Czech Science Foundation, and an APIF grant from the University of Barcelona.
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Albuquerque, H., Font, J.M., Jansana, R., Moraschini, T. (2018). Assertional logics, truth-equational logics, and the hierarchies of abstract algebraic logic. In: Czelakowski, J. (eds) Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science. Outstanding Contributions to Logic, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-319-74772-9_2
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